In modulation theory, reliance on a modulational Ansatz implies that an additional step is required to reconstruct
physical quantities such as the surface elevation from the solution of the envelope equation.
This step is determined as part of the asymptotic analysis and, in classical approaches like the method of multiple scales,
this reconstruction is typically carried out perturbatively via a Stokes-type expansion.
It is an important computation that also influences the model's performance,
as shown e.g. in comparisons with experimental data on extreme waves [1]}.
In the present context, the reconstruction procedure is associated with
the Birkhoff normal form transformation to eliminate all non-resonant triads.
It is a non-perturbative computation but requires solving an auxiliary system of PDEs for the surface variables.
In this way, higher-order harmonic contributions to the wave spectrum are automatically generated from
the carrier wave component through nonlinear interactions according to these PDEs.
We provide a detailed derivation of this auxiliary system for nonzero constant vorticity,
noting that the resulting equations are significantly more complicated than in the irrotational case.
By definition, these equations also take the form of a Hamiltonian evolutionary system with a canonical symplectic structure.
As a consequence, the entire solution process fits within a Hamiltonian framework.
Based on this approximation, we conduct a linear stability analysis and examine the dependence on vorticity.
We then test analytical and numerical predictions from our model against direct numerical simulations of the full system,
by inspecting the time evolution of perturbed Stokes waves and their possible instability.
We also compare our results to numerical solutions of the related Dysthe model by [2]}
and discuss the performance of the different reconstruction methods.
Previous observations on the focusing (resp. defocusing) effects of negative (resp. positive) vorticity
associated with a counter-propagating (resp. co-propagating) current are recovered.
